Global wellposedness of NLS in $H^1(\mathbb{R}) + H^s(\mathbb{T})$

TL;DR

This paper proves global well-posedness for certain defocusing nonlinear Schrödinger equations in hybrid function spaces combining real line and torus components, extending previous local and global results.

Contribution

It establishes the global well-posedness of defocusing NLS in mixed Sobolev spaces, filling gaps between known local and global results for these equations.

Findings

Global well-posedness in $H^1( ext{R}) + H^{3/2+}( ext{T})$ for cubic NLS

Global well-posedness in $H^1( ext{R}) + H^{5/2+}( ext{T})$ for polynomial NLS

Complements existing local and global results in hybrid function space settings

Abstract

We show global wellposedness for the defocusing cubic nonlinear Schr\"odinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.